Triaxial Compression on Semi-solid Alloys

Pressurized metal casting processes (e.g., high-pressure die casting and squeeze casting) and processes that combine deformation processing with solidification processing (e.g., twin-roll casting and semi-solid forging) have the potential to be the most cost-effective and least environmentally damaging metal processing routes. Each of these processes applies multi-axial compression to the mushy zone, and the rheological response determines how the semi-solid material deforms in the die and/or how defects such as deformation-induced macrosegregation occur. To optimize these processes, it is necessary to have a predictive understanding of the pressure-dependent rheology of semi-solid alloys at solid fractions where a solid network is present, for semi-solid morphologies ranging from equiaxed dendritic to globular (Figure 1).

Past research has shown that compressive deformation of semi-solid alloys can lead to different types of behavior. The solid network can be compressed expelling liquid[1‐3] as a consequence both of grain rearrangement and grain deformation. The network can also exhibit Reynolds’ dilatancy[4] (also known as shear-induced dilation) where the grains push or lever one another apart as they begin to rearrange (translate and rotate) under load and extra liquid is drawn into the expanding interstices between grains.[5‐8] Since the solid network can contract or dilate, causing liquid to be expelled or drawn-in, volumetric strains can be important during the compressive deformation of semi-solid alloys. As the solid fraction increases within the solid network (Figure 1), deformation of the grains is thought to become an increasing component of deformation and constitutive models often treat the rheology as viscoplastic deformation of a strain-rate-sensitive partially cohesive porous solid skeleton saturated with liquid (e.g., [3,9]). There is also a complex coupling between the rearrangement and/or deformation of the solid and flow of the liquid within the low permeability mush.[10,11]

Good constitutive models that capture such volumetric strains and the pressure-dependent rheology, and link between the low solid fraction suspension regime and high solid fraction regime remain a significant challenge, and this area is under continuous development.[9,12] The tests required to support the development of such models are non-trivial, and one approach to expedite modeling capabilities is to look for analogue/similar materials and assess whether models or general frameworks that describe their behavior under different conditions can be used or adapted for use with semi-solid metals.

In previous publications, some similarities in the response to loading have been noted between semi-solid alloys and granular soils,[2,5,13] and a range of mechanical testing procedures similar to those developed to study soil mechanics have been applied to semi-solid alloys, including vane tests,[14,15] direct shear tests,[16,17] confined uniaxial filtration,[2,3] and triaxial compression.[18‐20] However, while these tests are similar to tests procedures used in soil mechanics, unfortunately, they lack some important features necessary to enable a comprehensive understanding of the material behavior. For example, while the direct shear test devices used in soil mechanics can apply a normal stress and measure volumetric strains, these features have been omitted in direct shear testing on semi-solid alloys.[16,17] The most common approach to measuring semi-solid deformation has been unconfined compression (i.e., parallel plate compression),[1,11,21‐28] but these tests usually lack a method to link the volumetric behavior to the applied stress regime. In one study that did measure volume changes during parallel plate compression,[29] it was found that volumetric strains can be an important part of the semi-solid mechanical response.

Triaxial compression tests on semi-solid alloys where the sample is subjected to controlled pressure are limited in the literature. Some pioneering triaxial compression tests on semi-solid alloys have been carried out by Martin et al.[2,18] and Nguyen et al.[19,20] on semi-solid Sn-15 wt pct Pb. However, these tests had no means of continuous measurement of the sample interstitial liquid change during tests. Using this configuration, sample contraction could be studied by the expulsion of liquid, however, it was impossible to quantify or detect any increase in sample volume by liquid in-flow during shear-induced dilation.

This paper describes the design and development of a novel triaxial setting for semi-solid alloy testing. The design aim was to enable acquisition of data which link the volumetric deformation behavior with the applied stress regime in a semi-solid alloy at a range of confining pressures. The resultant rig allows measurement of specimen volumetric change due to either the in-flow or expulsion of liquid from the sample. Specimens of globular semi-solid Al-15 wt pct Cu alloy samples were tested to obtain such data and to evaluate the potential use of soil mechanics deformation theories for mushy zone behavior interpretation.

Since the triaxial compression method used here is similar to that widely used in soil mechanics, it is useful to interpret the measured semi-solid alloy behavior in a soil mechanics framework and to compare the response between the two types of material.

Figure 9 shows the development of the angle of shearing resistance, ϕ’ which was calculated using equation (3), against axial strain. An ultimate (end-of-test) ϕ’ of around 20° for all tests can be noted, while sample 300 kPa-549 (which underwent shear-induced dilation, Figure 6) exhibited a maximum angle of shearing resistance of about 25° at around 5 pct axial strain before falling to reach the ultimate value. The maximum (peak) angle of shearing resistance is associated with the point of maximum dilation (Figure 6(b)), similar to past work on shearing of dense soil samples.[32,38] These values of \( \phi^{\prime} \) are relatively low compared with values measured for sands (between 27° and 33° for ultimate and 35° to 45° for maximum angle of shearing[38]). However, data for almost spherical relatively smooth glass ballotini samples tested in a soil mechanics triaxial cell give an ultimate \( \phi^{\prime} \) of 21°.[39] These moderately low values of angle of shearing are likely to be because the 8-day semi-solid coarsening used in this work, and further time during semi-solid deformation, produced particularly smooth globules (Figure 4) which will give a low friction coefficient.

Extensive experimental research on both cohesive soils (clays) and cohesionless soils (sands) has established that, irrespective of the initial packing density or stress state, when a given soil is subject to shear deformation, the packing density and stress state can be described by a unique relationship in e:q:p′ space termed the Critical State Locus or line (CSL). A three-dimensional perspective of the CSL is presented in Figure 10(a).[40] For simplicity, this relationship is often presented using two (two-dimensional) projections. The first projection maps the CSL onto the p′–q plane, as is illustrated in yellow on Figure 10(a). This projection is also presented in Figure 10(c) and it is clear that from this perspective the CSL plots as a straight line. The second projection maps the CSL onto the e- p’ plane which is illustrated in blue on Figure 10(a). It is clear from Figure 10(a) that the projection onto this plane is a curve. However, in soil mechanics, this projection is most often plotted on a semi-logarithmic scale (e–log p′) as in Figure 10(b); using these axes it appears as a straight line. Figure 10(b) shows also a typical path of isotropic compression of a sample (represented by gray curve), at pressures lower than the material yield pressure when it joins what is called the Isotropic Normal Compression Line (Iso NCL, shown in Figure 10(b) by the broken black line). The Iso NCL is a unique line for the material that all isotropically compressed samples will join eventually when they are compressed beyond their yield pressure. Within the Critical State Soil Mechanics (CSSM) framework, the initial “state” of the material, i.e., the initial values of e and p′ and their position relative to the CSL for the material under consideration, determines the mechanical behavior. Upon the application of shear loading or deformation, samples with combined e and p′ values that lie above the CSL, e.g., point A shown in Figures 10(a) and (b), will contract (the grains will move to be closer to each other and liquid will be expelled), producing a denser sample with lower void ratio (higher solid fraction) at a point such as A’. When the e–ln p′ combination is below the CSL, as is the case for point B (Figures 10(a) and (b)), the sample will dilate during shearing (the solid particles will move apart from each other and liquid will be drawn in) reaching failure or steady state at a point such as B’ where the sample has higher void ratio (lower solid fraction); i.e., both e and p′ determine whether a soil is considered to be loose or dense. The peak (maximum) angle of shearing resistance (\( \phi^{\prime}_{\text{peak}} \)) is mobilized during dilation of dense samples. In granular materials, if two samples of different densities are subject to triaxial shearing at the same effective confining pressure (\( \sigma^{\prime}_{\text{cell}} = \sigma^{\prime}_{ 3} \)), the denser sample will mobilize a higher \( \phi^{\prime}_{\text{peak}} \) (as in Test 300 kPa-549 in Figure 9). The ultimate, or critical state, angle of shearing resistance (\( \phi^{\prime}_{\text{cs}} \)) is mobilized when a steady state is reached at which increasing the strain will not cause any increase in the stress. The value of \( \phi^{\prime}_{\text{cs}} \) is an intrinsic property of a soil or granular material which does not change with the initial states of different samples that can be created from the given material. In line with this overall trend of material behavior, the data on Figure 9 show that each sample mobilized approximately the same value \( \phi^{\prime} \) = \( \phi^{\prime}_{\text{cs}} \) of 20° at the end of testing.

The test series documented here considered a range of initial e values and applied cell pressures, \( \sigma^{\prime}_{\text{cell}} \), so that a CSL could be identified, enabling the data to be explored within the CSSM framework. For any drained triaxial compression test, the combination of stresses experienced by that particular sample during shearing (the stress path) will plot as a straight line on the \( p^{\prime} - q \) plane with a gradient of 3:1, as shown in Figure 10(c). (This gradient can be derived by expressing \( \sigma^{\prime}_{1} \) in Equation 2 in terms of \( q \) and \( \sigma^{\prime}_{\text{cell}} \)). The schematics on Figure 10(c) indicate that the stress path for a sample such as A will approach the CSL monotonically. However, in the case of a dense sample, such as B, the stress path will extend beyond the CSL as the peak deviator stress is mobilized, returning to the CSL at large strains. Figure 11(a) presents the stress paths for the four tests completed in the current study. Using this approach to present the data, the CSL is taken as the locus of points that give the \( p^{\prime} \) and \( q \) values at the end of each tests, as shown in Figure 11(a). Just like sample B on Figure 10(c), the stress path for test 300 kPa-549 extends above the CSL, as it mobilizes its maximum deviator stress before it returns along the same linear path to end close to the CSL; this is a response that is typical for dense, dilative soils.[32,38,41] The fact that a linear relationship between the end of tests points is observed here, indicates that this material conforms to our understanding of material behavior within the CSSM framework. The slope (\( M \)) of the CSL plotted on the \( p^{\prime} - q \) axes is directly related to \( \phi^{\prime}_{\text{cs}} \) as \( \sin \left( {\phi^{\prime}_{\text{cs}} } \right) = 3M /\left( {6 + M} \right) \). The \( M \) value for the tests described here is around (\( M \) = 0.8), reflecting the value of \( \phi^{\prime}_{\text{cs}} \) stated above. The ballotini samples in Reference 39 had \( M \) = 0.82.

The responses observed during the triaxial tests are plotted in the mean effective stress–void ratio (e–log p′) plane in Figure 11(b) using the semi-logarithmic axes typically employed in soil mechanics. The sample states at the end of the isotropic compression stage are indicated by the black circles in the figure. These points enable the compression behavior of the material to be described by the black line labeled “compression line” in Figure 11(b). Comparing this compression line with the typical behavior of soil during compression shown previously in Figure 10(b), e.g.,[42] it can be seen that the pressures reached in these tests of semi-solid alloy (black curve in Figure 11(b)) do not reach the yield pressure of the material, when the material compression follows the straight normal compression line (NCL), rather the data agree well with the curved gray part of the compression curve in Figure 10(b).

The black points defining the compression line on Figure 11(b) also represent the sample condition at the start of the shearing stage during the test. The thin gray dotted lines on Figure 11b) describe the e–p′ relationships during the shearing stage of the experiments. The data at the end of each test, when it is assumed that a critical state has been attained, are represented by the gray circles on Figure 11(b). These points lie approximately on a straight line (in the e-log p’ plane) consistent with the concept of a Critical State Line and adhering to the CSSM framework presented in Figure 10. The CSL in Figure 11(b) lies to the left of NCL on this plane, similar to Figure 10(b), again confirming that the material behavior conforms to the CSSM framework.

It can be seen from Figures 6, 9, and 11 that, for the globular semi-solid alloy at ~ 70 to 85 vol pct solid studied here, it is the combination of initial solid fraction, g_s (or void ratio, e) and initial mean effective pressure, p′, that determine whether the material will undergo shear-induced dilation or contraction, and knowledge of the location of a sample relative to the CSL in e:q:p′ space enables prediction of the response to shear. Samples whose initial state (i.e., following any compression state) is to the right of and above the CSL in the e–log p′ plane contract during shearing, experiencing a reduction in void ratio (increase in solid fraction) prior to reaching the CSL, as in samples 600 kPa–550 and 1200 kPa–550 in Figure 11(b). On the other hand, samples whose initial state is on the left of and below the CSL will exhibit some dilation (decrease in solid fraction) and mobilize a peak stress during shearing before reaching their CSL as in the case of sample 300 kPa–549 in Figure 11(b). This is consistent with the CSSM framework developed for soils e.g., References 32,43‐45.

The idea of applying soil mechanics theories to model the mushy zone has also been considered in the previous literature,[3,13,14,46] for example by the use of Terzaghi’s concept of effective stress[31] and the application of some laboratory testing procedures similar to those carried out on soils.[3,14,17,18,20,22,47] Further to this, in the present work, it can be deduced that globular semi-solid alloys at ~ 70 to 85 pct solid have a pressure sensitivity similar to soils both in terms of the pressure-dependent flow stress, and also a pressure-dependent volumetric response, and the concept of a critical state locus in e:q:p′ space seems to be both valid and useful for predicting the rheological response of semi-solid alloys, at least for the range of solid fraction and strain rate used.

While these experiments have shown the potential of interpreting and modeling semi-solid alloy deformation within a framework similar to CSSM, further features need to be added in future to generalize this framework for semi-solid alloys. For example, other microstructural parameters are needed in addition to the solid fraction (void ratio) such as the dendrite/grain envelope fraction and envelope shape, and the strain rate dependence of the CSL in semi-solid alloys needs to be considered.[46] The framework also needs to link to/from suspension flow at lower solid fraction and to/from the viscoplastic deformation of a solid skeleton saturated with liquid at higher solid fraction.[9]

This work has shown the value of developing drained triaxial compression apparatus with a liquid reservoir capable of measuring both the in-flow and out-flow of interstitial liquid during the isothermal deformation of semi-solid alloys. While these experiments are significantly more complex than techniques such as parallel plate compression and direct shear that have been widely applied to semi-solid alloys previously,[1,11,21,22,24‐27] this approach enables the changing solid fraction (or void ratio), deviator stress, and the mean effective pressure to be known during compression and shearing (Figures 11(a) and (c)), which are the key parameters in a critical state framework (Figure 9). Knowledge of the changing solid fraction during shearing is of additional importance for semi-solid alloys because, when liquid is expelled or drawn-in from a region of mush due to shear-induced contraction or dilation, it creates macrosegregation which affects the microstructure and properties of the casting, especially when strain localization occurs. Furthermore, knowledge of whether a local region will undergo shear-induced contraction or dilation provides knowledge of whether this form of deformation-induced macrosegregation will be negative or positive in this region. Such triaxial compression is likely to be relevant in, for example, the soft rolling reduction applied near the end of the sump in the continuous casting of steels,[48] and in twin-roll casting of Al and Mg alloys.[49]

A new drained triaxial compression apparatus based on Bishop’s triaxial apparatus used in mechanical testing of soil has been developed for semi-solid alloys. Drained triaxial compression experiments have been performed on globular semi-solid Al-15 wt pct Cu at 70 to 85 vol pct solid, with confining pressures from 300 to 1200 kPa. The results were then analyzed and compared with past work on soils in similar apparatus. The following conclusions can be drawn: